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      https://www.ias.ac.in/article/fulltext/pmsc/131/0019

    • Keywords

       

      Lattice points; irrationality measure; diophantine approximation.

    • Abstract

       

      For fixed positive real numbers $\omega , \omega '$, it is known that the number of lattice points $(u,v), u\ge 0, v \ge 0$ satisfying $0 \le u \omega + v\omega ' \le \eta $ is given by $\frac{1}{2}\big (\frac{\eta ^{2}}{\omega \omega ^{'}}+\frac{\eta }{\omega } +\frac{\eta }{\omega ^{'}}\big )+ O_{\varepsilon }(\eta ^{1-\frac{1}{\alpha _{0}} +\varepsilon })$, where $\alpha _0 \ge 1$ is a constant. In this paper, we explicitly compute $\alpha _0$ for certain values of $\omega /\omega '$. In particular, in Ramanujan's case (i.e., when $\omega = \log 2$ and $\omega ' = \log 3$), we show that $\alpha _0 = 2^{18}\log 3$ is admissible. This improves an earlier result of the paper (Ramachandra K, Sankaranarayanan A and Srinivas K, Hardy Ramanujan J. 19 (1996) 2--56), where it was shown that $\alpha _0 = 2^{40}\log 3$ holds.

    • Author Affiliations

       

      C G KARTHICK BABU1 USHA K SANGALE2

      1. Institute of Mathematical Sciences, HBNI, C.I.T. Campus, Taramani, Chennai 600 113, India
      2. SRTM University, Vishnupuri, Nanded 431 606, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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